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Theorem bnj964 29583
Description: Technical lemma for bnj69 29648. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj964.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj964.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj964.5  |-  ( ps'  <->  [. p  /  n ]. ps )
bnj964.8  |-  ( ps"  <->  [. G  / 
f ]. ps' )
bnj964.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj964.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj964.96  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
bnj964.165  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
Assertion
Ref Expression
bnj964  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ps" )
Distinct variable groups:    A, f,
i, n    D, i    i, G    R, f, i, n   
i, X    f, p, i    y, f, i, n   
i, m    ph, i
Allowed substitution hints:    ph( y, f, m, n, p)    ps( y, f, i, m, n, p)    ch( y, f, i, m, n, p)    A( y, m, p)    C( y,
f, i, m, n, p)    D( y, f, m, n, p)    R( y, m, p)    G( y, f, m, n, p)    X( y, f, m, n, p)    ps'( y, f, i, m, n, p)    ps"( y, f, i, m, n, p)

Proof of Theorem bnj964
StepHypRef Expression
1 nfv 1751 . . . 4  |-  F/ i ( R  FrSe  A  /\  X  e.  A
)
2 bnj964.2 . . . . . . . 8  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
32bnj1095 29422 . . . . . . 7  |-  ( ps 
->  A. i ps )
4 bnj964.3 . . . . . . 7  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
53, 4bnj1096 29423 . . . . . 6  |-  ( ch 
->  A. i ch )
65nfi 1670 . . . . 5  |-  F/ i ch
7 nfv 1751 . . . . 5  |-  F/ i  n  =  suc  m
8 nfv 1751 . . . . 5  |-  F/ i  p  =  suc  n
96, 7, 8nf3an 1985 . . . 4  |-  F/ i ( ch  /\  n  =  suc  m  /\  p  =  suc  n )
101, 9nfan 1983 . . 3  |-  F/ i ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )
11 bnj255 29339 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) )
12 bnj645 29389 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  suc  i  e.  p )
13 simp3 1007 . . . . . . . 8  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  p  =  suc  n )
1413bnj706 29393 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  p  =  suc  n )
15 eleq2 2493 . . . . . . . . 9  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
1615biimpac 488 . . . . . . . 8  |-  ( ( suc  i  e.  p  /\  p  =  suc  n )  ->  suc  i  e.  suc  n )
17 elsuci 5499 . . . . . . . . 9  |-  ( suc  i  e.  suc  n  ->  ( suc  i  e.  n  \/  suc  i  =  n ) )
18 eqcom 2429 . . . . . . . . . 10  |-  ( suc  i  =  n  <->  n  =  suc  i )
1918orbi2i 521 . . . . . . . . 9  |-  ( ( suc  i  e.  n  \/  suc  i  =  n )  <->  ( suc  i  e.  n  \/  n  =  suc  i ) )
2017, 19sylib 199 . . . . . . . 8  |-  ( suc  i  e.  suc  n  ->  ( suc  i  e.  n  \/  n  =  suc  i ) )
2116, 20syl 17 . . . . . . 7  |-  ( ( suc  i  e.  p  /\  p  =  suc  n )  ->  ( suc  i  e.  n  \/  n  =  suc  i ) )
2212, 14, 21syl2anc 665 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  ( suc  i  e.  n  \/  n  =  suc  i ) )
23 df-3an 984 . . . . . . . . . . . . 13  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
)  <->  ( ( i  e.  om  /\  suc  i  e.  p )  /\  suc  i  e.  n
) )
24233anbi3i 1198 . . . . . . . . . . . 12  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
( i  e.  om  /\ 
suc  i  e.  p
)  /\  suc  i  e.  n ) ) )
25 bnj255 29339 . . . . . . . . . . . 12  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  suc  i  e.  n )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
( i  e.  om  /\ 
suc  i  e.  p
)  /\  suc  i  e.  n ) ) )
2624, 25bitr4i 255 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  suc  i  e.  n ) )
27 bnj345 29348 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  suc  i  e.  n )  <->  ( suc  i  e.  n  /\  ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  ( i  e.  om  /\ 
suc  i  e.  p
) ) )
28 bnj252 29337 . . . . . . . . . . 11  |-  ( ( suc  i  e.  n  /\  ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) )  <->  ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
2926, 27, 283bitri 274 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  <->  ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
3011anbi2i 698 . . . . . . . . . 10  |-  ( ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) )  <->  ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
3129, 30bitr4i 255 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  <->  ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) ) )
32 bnj964.96 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
3331, 32sylbir 216 . . . . . . . 8  |-  ( ( suc  i  e.  n  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
3433ex 435 . . . . . . 7  |-  ( suc  i  e.  n  -> 
( ( ( R 
FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
35 df-3an 984 . . . . . . . . . . . . 13  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  <->  ( (
i  e.  om  /\  suc  i  e.  p
)  /\  n  =  suc  i ) )
36353anbi3i 1198 . . . . . . . . . . . 12  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
( i  e.  om  /\ 
suc  i  e.  p
)  /\  n  =  suc  i ) ) )
37 bnj255 29339 . . . . . . . . . . . 12  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  n  =  suc  i )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
( i  e.  om  /\ 
suc  i  e.  p
)  /\  n  =  suc  i ) ) )
3836, 37bitr4i 255 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  <->  ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  n  =  suc  i ) )
39 bnj345 29348 . . . . . . . . . . 11  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
)  /\  n  =  suc  i )  <->  ( n  =  suc  i  /\  ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) )
40 bnj252 29337 . . . . . . . . . . 11  |-  ( ( n  =  suc  i  /\  ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) )  <->  ( n  =  suc  i  /\  (
( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
4138, 39, 403bitri 274 . . . . . . . . . 10  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  <->  ( n  =  suc  i  /\  (
( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
4211anbi2i 698 . . . . . . . . . 10  |-  ( ( n  =  suc  i  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) )  <->  ( n  =  suc  i  /\  (
( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) ) ) )
4341, 42bitr4i 255 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  <->  ( n  =  suc  i  /\  (
( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) ) )
44 bnj964.165 . . . . . . . . 9  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
4543, 44sylbir 216 . . . . . . . 8  |-  ( ( n  =  suc  i  /\  ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p ) )  -> 
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) )
4645ex 435 . . . . . . 7  |-  ( n  =  suc  i  -> 
( ( ( R 
FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
4734, 46jaoi 380 . . . . . 6  |-  ( ( suc  i  e.  n  \/  n  =  suc  i )  ->  (
( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
4822, 47mpcom 37 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  i  e.  om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
4911, 48sylbir 216 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p
) )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) )
50493expia 1207 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  -> 
( ( i  e. 
om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
5110, 50alrimi 1927 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  A. i ( ( i  e.  om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R ) ) )
52 bnj964.5 . . . . 5  |-  ( ps'  <->  [. p  /  n ]. ps )
53 vex 3081 . . . . 5  |-  p  e. 
_V
542, 52, 53bnj539 29531 . . . 4  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  p  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
55 bnj964.8 . . . 4  |-  ( ps"  <->  [. G  / 
f ]. ps' )
56 bnj964.12 . . . 4  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
57 bnj964.13 . . . 4  |-  G  =  ( f  u.  { <. n ,  C >. } )
5854, 55, 56, 57bnj965 29582 . . 3  |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  p  ->  ( G `
 suc  i )  =  U_ y  e.  ( G `  i ) 
pred ( y ,  A ,  R ) ) )
5958bnj115 29360 . 2  |-  ( ps"  <->  A. i
( ( i  e. 
om  /\  suc  i  e.  p )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R ) ) )
6051, 59sylibr 215 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  ps" )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1867   A.wral 2773   [.wsbc 3296    u. cun 3431   {csn 3993   <.cop 3999   U_ciun 4293   suc csuc 5435    Fn wfn 5587   ` cfv 5592   omcom 6697    /\ w-bnj17 29320    predc-bnj14 29322    FrSe w-bnj15 29326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-suc 5439  df-iota 5556  df-fv 5600  df-bnj17 29321
This theorem is referenced by:  bnj910  29588
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